What Are Classical Lie Algebras A, B, C, and D?

In summary, classical Lie algebras A, B, C, and D refer to specific types of Lie algebras that arise from the study of symmetries and are associated with certain root systems. Algebra A_n corresponds to the special linear group SL(n+1), characterized by trace-zero matrices. Algebra B_n relates to the orthogonal group O(n), dealing with symmetries of n-dimensional space. Algebra C_n is associated with the symplectic group Sp(2n), relevant in Hamiltonian mechanics. Finally, algebra D_n corresponds to the even orthogonal group, reflecting symmetries in even-dimensional spaces. Each of these algebras has unique properties and applications in mathematics and theoretical physics.
  • #1
HDB1
77
7
TL;DR Summary
..
Dear @fresh_42 , Hope you are well.

Please, I have a question if you do not mind, about Lie Algebra,

In page 2 in the book of Lie algebra, written by Humphreys,

Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal..

Could you please explain with simple example of them.

Sorry for bothering you, and thank you so much in advance,
 
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  • #2
HDB1 said:
TL;DR Summary: ..

Dear @fresh_42 , Hope you are well.

Please, I have a question if you do not mind, about Lie Algebra,

In page 2 in the book of Lie algebra, written by Humphreys,

Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal..

Could you please explain with simple example of them.

Sorry for bothering you, and thank you so much in advance,
You are asking him to write a book chapter! You need to be more specific. Did you get type A? What exactly did you not understand about B, C and D?
 
  • #3
Dear @martinbn,

No, classical lie algebras A,B,C, and D are not chapters, they are definitions in page number 2.

Thanks,
 
  • #4
HDB1 said:
Dear @martinbn,

No, classical lie algebras A,B,C, and D are not chapters, they are definitions in page number 2.

Thanks,
I know what they are! My point is that your question is too general.

I can ask you again. Did you understand type A? What exactly do you find unclear about types B, C and D?
 
  • #5
HDB1 said:
TL;DR Summary: ..

Dear @fresh_42 , Hope you are well.

Please, I have a question if you do not mind, about Lie Algebra,

In page 2 in the book of Lie algebra, written by Humphreys,

Classical Lie algebras, ##A, B, C## and ##D##, I did not get it well, especially, symplectic and orthogonal..

Could you please explain with simple example of them.

Sorry for bothering you, and thank you so much in advance,
He has already written an insights article about this. Have you read it? https://www.physicsforums.com/insig...h-the-basics/#3-Classical-simple-Lie-Algebras
 

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